1. Parameterizations in Data Assimilation Philippe Lopez Physical Aspects Section, Research Department, ECMWF (Room 113) ECMWF Training Course 10-12 May 2006

2. Parameterizations in Data Assimilation Introduction Two examples of physical initialization A very simple assimilation problem 3D-Var assimilation The concept of adjoint 4D-Var assimilation Tangent-linear and adjoint coding Issues related to physical parameterizations in assimilation Physical parameterizations in ECMWF’s current 4D-Var system Recent developments for assimilation in precipitation regions at ECMWF Conclusions

3. General features of data assimilation  Goal: to produce an accurate four dimensional representation of the atmospheric state to initialize numerical weather prediction models.  This is achieved by combining in an optimal statistical way all the information on the atmosphere, available over a selected time window (usually 6 or 12 hours): Observations with their accuracies (error statistics), Short-range model forecast (background) with associated error statistics, Atmospheric equilibriums (e.g. geostrophic balance), Physical laws (e.g. perfect gas law, condensation)  The optimal atmospheric state found is called the analysis.

4. Which observations are assimilated? Operationally assimilated since many years ago: * Surface measurements (SYNOP, SHIPS, DRIBU,…), * Vertical soundings (TEMP, PILOT, AIREP, wind profilers,…), * Geostationary satellites (METEOSAT, GOES,…) Polar orbiting satellites (NOAA, SSM/I, AIRS, AQUA, QuikSCAT,…): - radiances (infrared, passive microwave, visible), - products (motion vectors, total column water vapour, ozone,…). More recently: * Satellite radiances/retrievals in cloudy/precipitation regions, * Precipitation measurements from ground-based radars and gauges. Still experimental: * Satellite cloud/precipitation radar reflectivities/products, * Lidar backscattering/products (wind vectors, water vapour), * GPS water vapour retrievals, * Satellite measurements of aerosols, CO2,....

5. Why physical parameterizations in data assimilation?  In current operational systems, most used observations are directly or indirectly related to temperature, wind, surface pressure and humidity outside cloudy and precipitation areas.  Physical parameterizations are used during the assimilation to link the model’s prognostic variables (typically: T, u, v, q v and P s ) to the observed quantities (e.g., radiances, reflectivities,…).  Observations related to clouds and precipitation are starting to be routinely assimilated,  but how to convert such information into proper corrections of the model’s initial state (prognostic variables T, u, v, q v and, P s ) is not so obvious. For instance, problems in the assimilation can arise from the discontinuous or nonlinear nature of moist processes.

6. Improvements are still needed…  More observations are needed to improve the forecast of:  Mesoscale phenomena (convection, frontal regions),  Small-scale processes (planetary boundary layer and surface),  The tropical circulation (monsoons, squall lines, tropical cyclones).  Recent developments and improvements have been achieved in:  Data assimilation techniques (e.g. 3D-Var  4D-Var),  Physical parameterizations in NWP models (prognostic schemes, detailed convection and large-scale condensation processes),  Radiative transfer models (infrared and microwave frequencies),  Horizontal and vertical resolutions of NWP models (currently at ECMWF: T799 ~ 25 km, 91 vertical levels),  New satellite platforms and instruments (incl. microwave imagers, precipitation/cloud radars, lidars).

7. To summarize… Observations (y o ) with errors a priori information from model: Background field (x b ) with errors Data assimilation system (e.g., 4D-Var) Analysis (x a ) Physical parameterizations are needed: - to link the model state to the observed quantities, - to evolve the model state in time during the assimilation (e.g., 4D-Var).

8. Empirical Initialization Example 1: Ducrocq et al. (2000) - Using the mesoscale research model Méso-NH (prognostic clouds and precipitation). - Particular focus on strong convective events. - Method: Prior to the forecast: 1) A mesoscale surface analysis is performed (esp. to identify convective cold pools) 2) the model humidity, cloud and precipitation fields are empirically adjusted to match ground-based precipitation radar observations and METEOSAT infrared brightness temperatures. Radar METEOSAT

9. Study by Ducrocq et al. (2004) with 2.5-km resolution model Méso-NH Flash-flood over South of France (8-9 Sept 2002) + Nîmes + 12h FC from operational analysis + Rain gauges Nîmes radar + 12h FC from modified analysis 100 km 12h accumulated precipitation: 8 Sept 12 UTC  9 Sept 2002 00 UTC

10. Initialization with physical parameterization Example 2: Krishnamurti et al. (1988, 1993) Model state: x = (T, q, Div, Vor, Log(P s )) Set of surface rainfall observations: R obs Kuo’s convection scheme is inverted to modify the model state so as to match the surface rainfall observations: Newtonian relaxation (or nudging) procedure over one or two days: where N is the adjustable nudging coefficient and F denotes physical and advective processes. Newtonian relaxation Standard Forecast day-1day 0 time

11. Physical initialization (Krishnamurti et al. 1988, 1993) Moisture convergence: Heating and moistening due to moist convection: where b is a moistening parameter: b=f(Relative Humidity) with f(0)=1 and f(1)=0. Moisture conservation: I = R + M = (1-b)I + bI where R is the surface rainfall rate and M is the moistening of the environment. Kuo’s convection scheme:

12. Physical initialization (Krishnamurti et al. 1988, 1993) Given an observation of surface precipitation R obs, find q mod such that:  Inversion of the Kuo scheme (with b=0). which ensures conservation of total column water vapour: with

13. Physical initialization (Krishnamurti et al. 1988, 1993) Time evolution of globally averaged evaporation and precipitation for a case study starting at 1200 UTC 11 July 1979 (units mm day -1 ). These results have been obtained with the Florida State University (FSU) global spectral model: a)based on a straight run of the model from day 0 b)based on a one-day Newtonian relaxation on day –1 c)based on a two-day Newtonian relaxation started on day -2.

14. Physical initialization (Krishnamurti et al. 1993) Satellite IR image showing hurricanes Freda, Gerald and Holly over the western Pacific. 40N 30N 20N 10N Eq 4-day forecast of mean sea level pressure (in hPa) without (top) and with (bottom) 1-day physical initialization with SSM/I observed rainfall rates. 9 September 1987 0000 UTC 10 September 1987 1200 UTC

16. 2

18.  Minimizing the cost function J is equivalent to finding the so-called Best Linear Unbiased Estimator (BLUE), if:  Errors on the background and on the observations are assumed to be unbiased.  Background and observation errors are assumed to be uncorrelated.  If the statistical distributions of errors are Gaussian, then the final analysed state is also the maximum likelihood estimator of the true state.  The analysis is obtained by adding corrections to the background which depend linearly on background-observations departures.  In this linear context, the observation operator (to go from model space to observation space) must not be too non-linear in the vicinity of the model state, else the result of the analysis procedure is not optimal.  The result of the minimization depends on the background and observation error statistics (matrices B and R ) but also on the Jacobian of the observation operator H. Important remarks on variational data assimilation

19. Marécal and Mahfouf (2002) Betts-Miller (adjustment) Jacobians of surface rainfall rate w.r.t. T and q v Tiedtke (ECMWF’s oper mass flux) Example of observation operator : H = convection scheme model state (T,q v )  simulated surface rainfall rate

23. Finding the minimum of cost function J  iterative minimization procedure cost function J model variable x 2 model variable x 1 J(xb)J(xb) J mini

30. 70 35 T799 L91 T255 L91

31. …3 slides of summary about 3D-Var, 4D-Var and incremental 4D-Var

32. 9 time 1512 xbxb All observations y o between t a -3h and t a +3h are assumed to be valid at analysis time (t a =1200 UTC here) yoyo xaxa 3D-Var x b = model first-guess x a = final analysis analysis time t a model state

33. Model trajectory from first guess x b Model trajectory from corrected initial state time 1512 9 xbxb All observations y o between t a -9h and t a +3h are valid at their actual time yoyo xaxa analysis time t a 4D-Var 63 model state assimilation window Forecast model is involved in minimization

34. titi t0t0 time xixi x ib x0x0 x 0b model space x MsMs M observation space time titi t0t0 HixiHixi H(x ib ) y y oi didi HiHi t 0 +12h H non-linear observation operator H i tangent-linear observation operator M non-linear forecast model (full physics) M s tangent-linear model (simplified physics) y oj Incremental 4D-Var trajectory from first guess x 0b trajectory from corrected initial state

35. t1t1 t2t2 x0x0 time x1x1 x2x2 y1y1 y2y2

36.  Variational data assimilation relies on two essential assumptions:  Gaussian distributions of model background and observation errors,  Quasi-linearity of all operators involved ( H, M ).  Given some background fields and a very large set of asynchronous observations available within a certain time window (e.g. 6 or 12h-long), 4D-Var searches the statistically optimal initial model state x 0 that minimizes the cost function: J (x 0 ) = J b (x 0 ) + J o (HM(x 0 ))  The calculation of  x0 J requires the coding of tangent-linear and adjoint versions of the observation operator H and of the full nonlinear forecast model M (including physical parameterizations).  The tangent-linear and adjoint models M and M T are often based on a simplified version of the full nonlinear model M to reduce computational cost in the iterative minimization and to avoid nonlinearities. Summary

39. As an alternative to the matrix method, adjoint coding can be carried out using a line-by-line approach.

40. Tangent linear codeAdjoint code δx = 0δx * = 0 δx = A δy + B δzδy * = δy * + A δx * δz * = δz * + B δx * δx * = 0 δx = A δx + B δzδz * = δz * + B δx * δx * = A δx * do k = 1, N δx(k) = A δx(k  1) + B δy(k) end do do k = N, 1,  1 (Reverse the loop!) δx * (k  1) = δx * (k  1) + A δx * (k) δy * (k ) = δy * (k) + B δx * (k) δx * (k) = 0 end do if (condition) tangent linear codeif (condition) adjoint code Basic rules for line-by-line adjoint coding (1) And do not forget to initialize local adjoint variables to zero ! Adjoint statements are derived from tangent linear ones in a reversed order

41. Tangent linear codeTrajectory and adjoint code if (x > x0) then δx = A δx / x x = A Log(x) end if ------------- Trajectory ---------------- x store = x (storage for use in adjoint) if (x > x0) then x = A Log(x) end if --------------- Adjoint ------------------ if (x store > x0) then δx * = A δx * / x store end if Basic rules for line-by-line adjoint coding (2) The most common sources of error in adjoint coding are: 1)Pure coding errors, 2)Forgotten initialization of local adjoint variables to zero, 3)Mismatching trajectories in tangent linear and adjoint (even slightly), 4)Bad identification of trajectory updates: To save memory, the trajectory can be recomputed just before the adjoint calculations.

42. machine precision reached Perturbation scaling factor

46. Issues related to the use of physical parameterizations in variational data assimilation Variational assimilation is based on the strong assumption that the analysis is performed in a linear framework. Weakly nonlinear observation operators can be linearized to match this condition. However, in the case of physical processes, strong nonlinearities can occur sometimes in the presence of discontinuous/non-differentiable processes (e.g. switches or thresholds in cloud water and precipitation formation). x y 0  x ( finite size perturbation)  y (nonlinear)  y (tangent-linear)

48. Janisková et al. 1999

49. Evolution of temperature increments (24-hour forecast) with the tangent linear model using different approaches for the exchange coefficient K in the vertical diffusion scheme. Perturbations of K included in TL Perturbations of K set to zero in TL

50. ECMWF’s linearized simplified physical package Five linearized simplified physical parameterizations are currently used in ECMWF’s 4D-Var (main simplifications compared to the non-linear schemes are written in red): Vertical diffusion: turbulent mixing in the surface and planetary boundary layers. - based on K-theory and Blackadar mixing length, - exchange coefficients = f(Richardson number) (Louis et al. 1982), - perturbations of exchange coefficients are neglected. Gravity wave drag: subgrid-scale orographic effects (Lott and Miller 1997). - only low-level blocking part is used. Large-scale condensation: adjustment to saturation to produce stratiform precipit. - melting of snow inside layers where T becomes > 2 o C, - precipitation evaporation reduced or even set to zero. Convection: mass flux approach (Tiedtke 1989) - only deep convection is considered, - perturbations of mass-flux are neglected, - detrainment of cloud properties in environment is disregarded. Radiation: TL and AD for longwave radiation only. - calculations in the troposphere only, - constant emissivity formulation, - no dependence of radiation on cloud fraction.

51. Evaluation of the tangent-linear approximation Time evolution of analysis increments  x 0 with a T63 L31 version of the ECMWF forecast model over a 24-hour period (15 March 1999).  Nonlinear model with full physics: M(x 0 +  x 0 )- M(x 0 )  Tangent-linear model adiabatic (no physics): M adiab (  x 0 )  Tangent-linear model with physics: M phys (  x 0 )  Tangent-linear approximation absolute error :  = | M(x 0 +  x 0 )- M(x 0 ) - M(  x 0 ) | If the inclusion of the physics in the tangent-linear model improves the fit to the nonlinear model, then  should be reduced. Physical parameterizations are introduced one by one.

52. Zonal cross-sections of changes in mean absolute errors [M(x 0 +  x 0 )- M(x 0 )-M(  x 0 )] when physical parameterizations are included (T63 L31) + vertical diffusion + vert. diff. + gravity wave drag + large-scale condensation + convection

53. Parameterizations of moist processes (convection, large-scale condensation and microwave radiative transfer) are key components of the assimilation of such data. Recent developments have led to more detailed parameterizations used in tangent- linear and adjoint computations:  New simplified convection scheme (Lopez and Moreau 2005)  New simplified cloud scheme (Tompkins and Janisková 2004)  Microwave Radiative Transfer Model (Bauer 2002, Moreau 2003) Undergoing work is aiming at replacing the indirect ‘1D-Var + 4D-Var’ method by a direct assimilation of SSM/I TBs in 4D-Var (like in clear-sky regions). Example: Use of parameterizations for the assimilation of SSM/I rainy brightness temperatures in ECMWF’s 4D-Var Until June 2005, the Special Sensor Microwave/Imager (SSM/I) brightness temperatures (TBs) were directly assimilated in ECMWF’s operational 4D-Var system only in clear-sky regions. Since June 2005, 19 and 22 GHz SSM/I TBs have also been assimilated in regions affected by precipitation and clouds using an indirect 2-step approach (Mahfouf and Marécal 2002, 2003), the so-called ‘1D+4D-Var’ method (~50,000 additional observations at each cycle).

54. ‘1D+4D-Var’ assimilation of SSM/I rainy brightness temperatures 1D-Var Background T,q v Interpolation onto model grid 19 and 22 GHz SSM/I TBs Moist physics Microwave Radiative Transfer Model Simulated TBs Increments: δT, δq v Assumption: δT<<δq v Step 1

55. ‘1D+4D-Var’ assimilation of SSM/I rainy brightness temperatures 4D-Var All other observations (radiosondes, surface, satellite clear-sky radiances,…) Physical parameterizations Increments: δT, δq v, δP s, δu, δv Step 2 Operational Analysis Pseudo-observation from 1D-Var rain

56. Normalized RMS forecast errors: 24-hour forecasts of 850 hPa geopotential September 2004 (RMSE exp  RMSE cntrl ) / max(RMSE exp  RMSE cntrl ) EXP: CY29R2 with SSM/I rainy TBs assimilation CNTRL: CY29R2 without SSM/I rainy TBs assimilation ‘1D+4D-Var’ assimilation of SSM/I rainy brightness temperatures

57. Left: MTSAT infrared image of typhoon MATSA approaching Taiwan and China on 4 August 2005 at 0000 UTC. Right: 4D-Var moisture increments with rain assimilation (color shading; units in %), 900 hPa wind increments (white arrows) and surface pressure (isolines). ‘1D+4D-Var’ assimilation of SSM/I rainy brightness temperatures

58. 1D-Var on TRMM/Precipitation Radar data Tropical Cyclone Zoe (26 December 2002 @1200 UTC; Southwest Pacific) TRMM Precipitation Radar MODIS image TRMM-PR swath Cross-section

59. 1D-Var on TRMM/Precipitation Radar data Tropical Cyclone Zoe (26 December 2002 @1200 UTC) Vertical cross-section of rain rates (top, mm h -1 ) and reflectivities (bottom, dBZ): observed (left), background (middle), and analysed (right). Black isolines on right panels = 1D-Var specific humidity increments. 2A25 RainBackground Rain1D-Var Analysed Rain 2A25 ReflectivityBackground Reflect.1D-Var Analysed Reflect.

60.  Physical parameterizations have become important components in recent variational data assimilation systems.  However, special care must be taken when deriving their linearized versions (regularizations / simplifications), in order to eliminate the discontinuities and the non-differentiability of the physical processes they are supposed to represent.  This is particularly true for the assimilation of observations related to precipitation, clouds, and soil moisture, to which a lot of efforts are currently devoted.  Developments required when developing new simplified parameterizations for assimilation:  Evaluation in terms of Jacobians (i.e. sensitivities to input parameters),  Systematic validation against observations,  Comparison to the non-linear version used in forecast mode (trajectory),  Numerical tests for tangent-linear and adjoint codes for small perturbations,  Validity of the linear hypothesis for perturbations with larger size (typical of analysis increments). General conclusions

61. Useful References Variational data assimilation: Lorenc, A., 1986, Quarterly Journal of the Royal Meteorological Society, 112, 1177-1194. Courtier, P. et al., 1994, Quarterly Journal of the Royal Meteorological Society, 120, 1367-1387. Rabier, F. et al., 2000, Quarterly Journal of the Royal Meteorological Society, 126, 1143-1170. The adjoint technique: Errico, R.M., 1997, Bulletin of the American Meteorological Society, 78, 2577-2591. Tangent-linear approximation: Errico, R.M. et al., 1993, Tellus, 45A, 462-477. Errico, R.M., and K. Reader, 1999, Quarterly Journal of the Royal Meteorological Society, 125, 169-195. Janisková, M. et al., 1999, Monthly Weather Review, 127, 26-45. Mahfouf, J.-F., 1999, Tellus, 51A, 147-166. Physical parameterizations for data assimilation: Županski, D., and F. Mesinger, 1995, Monthly Weather Review, 123, 1112-1127. Mahfouf, J.-F., 1999, Tellus, 51A, 147-166. Janisková, M. et al., 1999, Quarterly Journal of the Royal Meteorological Society, 125, 2465-2485. Tompkins, A. M., and M. Janisková, 2004, Quarterly Journal of the Royal Meteorological Society, 130, 2495-2518. Lopez, P., and E. Moreau, 2005, Quarterly Journal of the Royal Meteorological Society, 131, 409-436. Microwave Radiative Transfer Model for data assimilation: Bauer, P., May 2002, Satellite Application Facility for Numerical Weather Prediction NWPSAF-EC-TR-005, 27 pages. Assimilation of observations affected by precipitation and/or clouds: Krishnamurti, T.N. et al., 1988, Monthly Weather Review, 116, 907-920. Krishnamurti, T.N. et al., 1993, Tellus, 45A, 247-269. Fillion, L., and R.M. Errico, 1997, Monthly Weather Review, 125, 2917-2942. Ducrocq, V., et al., 2000, Quarterly Journal of the Royal Meteorological Society, 126, 3041-3065. Hou, A. et al., 2000, Monthly Weather Review, 128, 509-537. Marécal, V., and. J.-F. Mahfouf, 2002, Monthly Weather Review, 130, 43-58. Marécal, V., and. J.-F. Mahfouf, 2003, Quarterly Journal of the Royal Meteorological Society, 129, 3137-3160. Moreau, E. et al., 2004, Quarterly Journal of the Royal Meteorological Society, 130, 827-852.